Adaptive fuzzy fault-tolerant output feedback control of uncertain nonlinear systems with actuator faults based on dynamic surface technique

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Abstract

This article develops an adaptive fuzzy control method for accommodating actuator faults in a class of unknown nonlinear systems with unmeasured states. The considered faults are modeled as both loss of effectiveness and lock-in-place. With the help of fuzzy logic systems to approximate the unknown nonlinear functions, a fuzzy adaptive observer is developed for estimating the unmeasured states. Combining the backstepping technique with the dynamic surface control (DSC) approach, a novel adaptive fuzzy faults-tolerant control approach is constructed. It is proved that the proposed control approach can guarantee that all the signals of the resulting closed-loop system are bounded and the tracking error between the system output and the reference signal converges to a small neighborhood of zero by appropriate choice of the design parameters. The simulation example and comparisons with the previous method are provided to show the effectiveness of the control approach.

Introduction

Fuzzy control methodologies have emerged in the past decades as promising ways to approach nonlinear control problems [1], [2], [3]. Many remarkable results on fuzzy state-feedback control and feedback output control design for uncertain nonlinear systems based on T–S fuzzy model have been obtained [4], [5], [6], [7], [8], [9]. However, the above mentioned control schemes are based on the methods of T–S fuzzy model, not fuzzy logic systems [10], and they also did not use the backstepping technique. In recent years, many approximation-based adaptive backstepping control approaches have been developed to deal with uncertain nonlinear strict-feedback systems via fuzzy logic systems, see for examples [11], [12], [13], [14], [15], [16], [17], [18], [19], [20] and the references therein. Adaptive fuzzy backstepping control approaches in [11], [12], [13], [14] are for single-input and single-output (SISO) nonlinear systems, and in [15], [16], [17] are for multiple-input and multiple-output (MIMO) nonlinear systems, while those in [18], [19], [20] are for SISO/MIMO nonlinear systems with immeasurable states. Adaptive fuzzy backstepping control approaches can provide a systematic methodology of solving control problems for a larger class of unknown nonlinear systems, where fuzzy logic systems are used to approximate unknown nonlinear functions, and the backstepping design technique is applied to construct adaptive controllers and the adaptation parameter laws. The main features of the above adaptive approaches are as follows: (i) they can be used to deal with those nonlinear systems without satisfying the so called the matching condition, and (ii) they do not require the unknown nonlinear functions are linearly parameterized. Therefore, the approximator-based adaptive fuzzy backstepping control becomes one of the most popular design approaches to a large class of uncertain nonlinear systems.

Although a great development has been achieved for the adaptive backstepping control, the aforementioned control approaches assume that all the components of the considered nonlinear systems are in good operating conditions. As we know, some faults, such as actuators and sensors usually exist in many real processes, which can degrade the control performances and even result in the instability of the control system or even catastrophic accidents [21], [22], [23], [24], [25], [26], [27], [28]. The research on accommodating such failures and maintaining acceptable system performance is particularly important.

To handle the problem of nonlinear system with actuator or sensor faults, many fault-tolerant control (FTC) approaches have been developed, see for examples [29], [30], [31], [32], [33], [34] and the references therein. Ye and Yang [29], [30] presented adaptive fault-tolerant control for linear systems with both loss of effectiveness and lock-in-place actuator faults. Tao et al. and Tang et al. [31], [32] developed adaptive fault-tolerant controllers for a class of SISO nonlinear systems and MIMO nonlinear systems with the same actuator faults as in [29], [30], while [33], [34] developed observer-based adaptive backstepping fault-tolerant control approaches for some nonlinear systems with additive profile faults. However, the above mentioned fault-tolerant control schemes require that the considered nonlinear systems with the matching conditions or the nonlinear functions are known. To remove these limitations, authors in [35], [36] investigated a class of unknown SISO nonlinear strict-feedback systems with both loss of effectiveness and lock-in-place actuator faults, in which fuzzy logic systems are employed to approximate the unknown functions, and based on the backstepping technique, two adaptive fuzzy backstepping FTC schemes were developed. The proposed control schemes guarantee not only the stability, but also the robust performance of the failed system. On the basis of the results of [35], [36], authors in [37] proposed an adaptive fuzzy backstepping FTC scheme for unknown MIMO strict-feedback nonlinear systems, and the stability of the control system was given. The main limitation in [35], [36] is that the states of the systems are required to be measurable. Thus they cannot be applied to solve the problem of those nonlinear systems with the immeasurable states. As stated in [18], [19], [20], [38], in practice, state variables are often unmeasured for many nonlinear systems, which is very important in both theory and real world applications. Recently, an adaptive fuzzy output feedback FTC scheme has been developed in [39] for a class of uncertain SISO nonlinear systems with actuator failures by designing a state observer to estimate the unmeasured states. However, the proposed adaptive fuzzy output feedback FTC scheme suffer from the problem of “explosion of complexity”, which caused by differentiations of some nonlinear functions at each step within the conventional backstepping technique [40], [41], [42]. As a result, the complexity of the controller drastically grows as the order of the system increase.

It is worth pointing out that the problems of actuator or sensor faults and unmeasured states widely exists in the complex nonlinear practice systems, for example, induction motor systems, hypersonic vehicle systems, chemical process systems, and power systems. Therefore, the failure compensation control designs for nonlinear systems with unmeasured states and actuator or sensor faults is an important issue, thus motivated us for this study.

Motivated by the aforementioned observations, in this paper, an adaptive fuzzy fault-tolerant control method is developed for a class of unknown MISO nonlinear systems with the actuator faults of both the loss of effectiveness and lock-in-place, and without assuming that the states are available for measurement. With the help of fuzzy logic systems to approximate the unknown nonlinear functions, a fuzzy adaptive observer is developed to estimate the immeasurable states. By combining the backstepping technique with the DSC approach, a novel adaptive fuzzy fault-tolerant scheme is constructed. The main advantages of the proposed FTC scheme are as follows: (i) by designing a state observer, the proposed FTC method remove the restrictive assumption in [35], [36], [37] that all the states of the system be measured directly; and (ii) the proposed FTC method can overcome the problem of “explosion of complexity” inherent in [34], [35], [36], [37]. It is proved that the proposed FTC method can guarantee that all the signals of the resulting closed-loop system are bounded, and the tracking error converges to a small neighborhood of zero.

Section snippets

Nonlinear system descriptions

Consider the following MISO strict-feedback nonlinear system with actuator failures:{ẋ1=x2+f1(x1)+d1(t)ẋ2=x3+f2(X2)+d2(t)ẋn1=xn+fn1(Xn1)+dn1(t)ẋn=ϖTu+fn(Xn)+dn(t)y=x1where Xi=[x1,x2,,xi]TΩi with Ωi is a sufficient large compact set in Ri, i=1,2,,n; u=[u1,u2,,um]TRm is the input vector whose component may fail during the system operation, yR is the system output, fi, i=1,2,,n are unknown continuous nonlinear functions, ϖ=(ϖ1,ϖ2,,ϖm)TRm, ϖj, j=1,2,,m are known constant vectors,

Fuzzy state observer design

Note that the states x2,,xn in the system (1) are not available for measurement, thus a suitable state observer should be established to obtain the estimates them.

To this end, rewrite Eq. (1) as follows:Ẋ=AX+Ky+i=1nBi[fi(Xi)+di]+BϖTu=AX+Ky+i=1nBi[fi(X^i)+Δfi+di]+BϖTuwhereX^=(x^1,x^2,,x^n)T,K=[k1,k2,,kn]T,Bi=[010]TB=[001]T,Δfi=fi(Xi)fi(X^i),A=A=[k1Ikn00]Choose the vector K such that A is a Hurwitz matrix. Thus, given a positive definite matrix Q=QT>0, there exists a positive

FTC design and stability analysis

In this section, an adaptive fuzzy FTC scheme will be developed by using the backstepping technique and DSC approach, and the stability of the closed-loop system will be given.

According to [35], [36], the following control structure is adopted:vj=bj(X^n)u0where 0b̲jbj(X^n)b¯j, x¯nΩnRn, j=1,2,,m; b̲j and b¯j are the lower and upper bounds of bj(X^n), respectively. u0 is the designed adaptive fuzzy controller to be designed by the backstepping recursive design as follows.

Define the changes

Simulation study

In this section, an example is given to show the effectiveness of the proposed adaptive fuzzy backstepping fault tolerant method.

Consider the following uncertain nonlinear system:ẋ1=x2+f1(x1)+d1(t)ẋ2=f2(x1,x2)+ϖ1u1+ϖ2u2+d2(t)y=x1where f1(x1)=x1e0.5x1, f2(x1,x2)=x1sin(x22), d1(t)=d2(t)=0.1, ϖ=[0.8,0.8]T. The tracking reference signal is chosen as yd(t)=sin(t).

Define fuzzy membership as follows:μF1l(x^1)=exp[(x^13+l)2/16],l=1,,5.μF2l(x^1,x^2)=exp[(x^13+l)2/4]×exp[(x^23+l)2/4,l=1,,5We

Conclusions

This article has developed an adaptive fuzzy FTC method for a class of unknown MISO nonlinear systems with actuator faults and with unmeasured states. The considered faults are modeled as both loss of effectiveness and lock-in-place. With the help of FLSs to approximate the unknown nonlinear functions, a fuzzy adaptive observer has been developed for estimating the unmeasured states. Combining the backstepping technique with DSC approach, a novel adaptive fuzzy FTC approach has been

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    This work was supported by the National Natural Science Foundation of China (Nos. 61074014, 61203008), Liaoning Innovative Research Team in University (LT2012013).

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